defpred S₁[ Point of S] means $1 in X;
deffunc H₁( Point of S) -> Point of (R_NormSpace_of_BoundedLinearOperators S,T) = diff f,$1;
consider F being PartFunc of S,(R_NormSpace_of_BoundedLinearOperators S,T) such that
A2: ( ( for x being Point of S holds
( x in dom F iff S₁[x] ) ) & ( for x being Point of S st x in dom F holds
F . x = H₁(x) ) ) from SEQ_1:sch 3();
take F ; :: thesis: ( dom F = X & ( for x being Point of S st x in X holds
F /. x = diff f,x ) )
for y being set st y in dom F holds
y in X by A2;
then A3: dom F c= X by TARSKI:def 3;

now

let y be set ; :: thesis: ( y in X implies y in dom F )
assume A4: y in X ; :: thesis: y in dom F
X is Subset of S by A1, Th35;
hence y in dom F by A2, A4; :: thesis: verum

end;

then X c= dom F by TARSKI:def 3;
hence dom F = X by A3, XBOOLE_0:def 10; :: thesis: for x being Point of S st x in X holds
F /. x = diff f,x

now

let x be Point of S; :: thesis: ( x in X implies F /. x = diff f,x )
assume x in X ; :: thesis: F /. x = diff f,x
then A5: x in dom F by A2;
then F . x = diff f,x by A2;
hence F /. x = diff f,x by A5, PARTFUN1:def 8; :: thesis: verum

end;

hence for x being Point of S st x in X holds
F /. x = diff f,x ; :: thesis: verum